To solve the equation \( v(x) = x(8 - 2x)(10 - 2x) \), we will start by expanding the equation step by step. This will give us a polynomial form which makes it easier to analyze and solve.
Step 1: Expand the Expression
First, let's expand \( (8 - 2x)(10 - 2x) \).
\[ (8 - 2x)(10 - 2x) = 8 \cdot 10 - 8 \cdot 2x - 2x \cdot 10 + 2x \cdot 2x \]
Calculating each term:
\[ 8 \cdot 10 = 80 \] \[ -8 \cdot 2x = -16x \] \[ -2x \cdot 10 = -20x \] \[ 2x \cdot 2x = 4x^2 \]
Now combine these:
\[ (8 - 2x)(10 - 2x) = 80 - 16x - 20x + 4x^2 = 80 - 36x + 4x^2 \]
Step 2: Substitute Back
Now substitute this back into the original equation for \( v(x) \):
\[ v(x) = x(80 - 36x + 4x^2) \]
Distributing \( x \):
\[ v(x) = 80x - 36x^2 + 4x^3 \]
Step 3: Rearrange into Standard Polynomial Form
Rearranging the polynomial gives us:
\[ v(x) = 4x^3 - 36x^2 + 80x \]
Step 4: Set the Polynomial Equal to Zero
To solve for \( x \), set \( v(x) = 0 \):
\[ 4x^3 - 36x^2 + 80x = 0 \]
Step 5: Factor Out Common Terms
We can factor out \( 4x \):
\[ 4x(x^2 - 9x + 20) = 0 \]
Step 6: Solve Each Factor
The equation \( 4x = 0 \) gives:
\[ x = 0 \]
Next, we solve the quadratic \( x^2 - 9x + 20 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1, b = -9, c = 20 \):
\[ b^2 - 4ac = (-9)^2 - 4 \cdot 1 \cdot 20 = 81 - 80 = 1 \]
Now plug in the values into the formula:
\[ x = \frac{9 \pm \sqrt{1}}{2} \] \[ x = \frac{9 \pm 1}{2} \]
This gives us two potential solutions:
- \( x = \frac{10}{2} = 5 \)
- \( x = \frac{8}{2} = 4 \)
Step 7: Solutions
Thus, the solutions to the equation \( v(x) = 0 \) are:
\[ x = 0, \quad x = 4, \quad x = 5 \]
These are the points where the polynomial intersects the x-axis or where \( v(x) = 0 \).
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